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developmental transitions are controlled by the relative levels of dif-
ferent transcription factors (Wilkins, 1992; Davidson, 2001). Because
the control of most developmentally regulated genes is a consequence
of the synthesis of the factors that regulate their transcription, transi-
tions between cell types during development can be driven by changes
in the relative levels of a fairly small number of transcription factors
(Bateman, 1998). We can thus gain insight into the dynamical basis
of cell-type switching (i.e., differentiation) by focusing on molecular
circuits, or networks, consisting solely of transcription factors and
the genes that specify them. Such circuits, in which the components
mutually regulate one another™s expression, are termed autoregula-
tory. Cells make use of a variety of autoregulatory transcription-factor
circuits during development. It is obvious that such circuits, or net-
works, will have different properties depending on their particular
˜˜wiring diagrams,” that is, the patterns of interaction among their
components.
It is also important to recognize that transcription factors that
control cell differentiation not only initiate RNA synthesis on pro-
moters of template-competent genes (see above) but also are capable
of remodeling chromatin so as to transform their target genes from
the template-incompetent into the template-competent con¬guration
(Jimenez et al., 1992; Rupp et al., 2002).
Transcription factors can be classi¬ed as either activators, which
bind to a site on a gene™s promoter and enhance the rate of that
gene™s transcription over its basal rate, or repressors, which decrease
the rate of a gene™s transcription when bound to a site on its pro-
moter. The basal rate of transcription depends on constitutively active
transcription factors, which are distinct from those in the cell-type-
determining autoregulatory circuits that we will consider below. Re-
pression can be competitive or noncompetitive. In the ¬rst case, the
repressor will interfere with activator binding and can only depress
the gene™s transcription rate to the basal level. In the second case,
the repressor acts independently of any activator and can therefore
potentially depress the transcription rate below basal levels.
Keller (1995) used computational methods to investigate the be-
havior of several autoregulatory transcription-factor networks with a
range of wiring diagrams (Fig. 3.5). Each network was represented by
a set of n coupled differential equations -- one for the concentration
of each factor in the network -- and the steady-state behaviors of the
66 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO



XX XX
X
X
A B




XY
Y Y
C D

XY
XX




Y Y
E F

XX
X


Fig. 3.5 Six model genetic “circuits” encoding autoregulatory transcription factors.
(The diagrams represent circuits in the sense that there is feedback between their
components.) The green and blue horizontal bars represent the genes specifying the
transcription factors X (green symbols) and Y (blue symbols), respectively. The arrow
denotes the transcription start site. The portion of the gene to the left of the arrow is
the promoter. (A) Autoactivation by monomer X. (B) Autoactivation by dimer X2 . (C)
Mutual activation by monomer Y and dimer X2 . (D) Autoactivation by heterodimer XY.
(E) Mutual repression by monomers X and Y. (F) Mutual repression by dimer X2 and
heterodimer XY (with no binding site for the heterodimer on either promoter; see the
main text for details). Activating and repressing transcription factors are denoted by
circular and square symbols respectively.



systems were explored. The questions asked were the following. How
many stationary states exist? Are they stable or unstable?
To illustrate Keller™s approach, we discuss in more detail one such
network, designated as the ˜˜mutual repression by dimer and het-
erodimer” (MRDH) network, shown in Fig. 3.5F. It comprises the gene
encoding the transcriptional repressor X and the gene encoding the
protein Y, and thus represents a two-component network. Below we
summarize the salient points of the MRDH model.
1. The rate of synthesis of a transcription factor (i.e., d[X]/dt and
d[Y]/dt, [X] and [Y] being the respective concentrations of X and
Y) is proportional to the rate of transcription of the gene en-
coding that factor.
2. The transcription of a gene depends, in turn, on the spe-
ci¬c molecules that are bound to sites in the gene™s pro-
moter. Molecules can bind either as monomers (single protein
molecules, denoted by X, Y, etc.) or dimers (complexes of two
protein molecules). Both homodimeric complexes (X2 , Y2 ) and
heterodimeric complexes (XY) may be active. Thus a promoter
can be in various con¬gurations, with respective relative fre-
quencies, depending on its ˜˜occupancy,” that is, on how the
various potential factors bind to it. Speci¬cally, in the case of
the MRDH network, as characterized by Keller (Fig. 3.5F), the
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 67


promoter of gene X has no binding site for any molecule, either
activator or repressor: its only con¬guration is the empty state,
whose relative frequency therefore can be taken as 1. (Note that
the relative frequency of a promoter con¬guration is the prob-
ability of this con¬guration, thus the sum of all possible rela-
tive frequencies must be 1.) The (activator-independent or basal)
rate of synthesis of gene X is denoted by SXB . The promoter of
gene Y contains a single binding site for the dimeric form, X2 , of
the non-competitive repressor X. The monomeric forms of pro-
teins X and Y cannot bind DNA. Furthermore, proteins X and
Y can form a heterodimer XY that is also incapable of binding
DNA. Thus, while protein Y does not act directly as a transcrip-
tion factor, it affects transcription since it antagonizes the re-
pressor function of X by interfering with the formation of X2
(by forming XY). The promoter of gene Y can therefore be in two
con¬gurations: its binding site for X2 is either occupied or not
with respective relative frequencies K X K X2 [X]2 /(1 + K X K X2 [X]2 )
and 1/(1 + K X K X2 [X]2 ). (Note that the sum of these two relative
frequencies is 1.) Here K X and K X2 are the binding af¬nity of
X2 to the promoter of gene Y and the dimerization rate con-
stant for the formation of X2 , respectively, and we have used
the relationship K X [X2 ] = K X K X2 [X]2 . The synthesis of Y in both
con¬gurations is activator-independent and is denoted by SYB .
To incorporate the fact that X2 reduces the rate of transcription
of Y in a noncompetitive manner, in the occupied Y-promoter
con¬guration SYB is replaced by ρ SYB , with ρ ¤ 1.
3. The overall transcription rate of a gene is calculated as the sum
of products. Each term in the sum corresponds to a particular
promoter-occupancy con¬guration and is represented as a prod-
uct of two factors, namely, the frequency of that con¬guration
and the rate of synthesis resulting from that con¬guration. In
the MRDH network this rate for gene X is thus 1 — SXB (with SXB
as introduced above), because it has only a single promoter con-
¬guration. The promoter of gene Y can be in two con¬gurations
(with rates of synthesis SYB and ρ SYB , see above); therefore its
overall transcription rate is (1 + ρ K X K X2 [X]2 )SYB /(1 + K X K X2 [X]2 ).
4. The rate of decay of a transcription factor is a sum of terms,
each proportional to the concentration of a particular complex
in which the transcription factor participates. This is equiva-
lent to assuming exponential decay. For the transcriptional re-
pressor X in the MRDH network these complexes include the
monomer X, the homodimer X2 , and the heterodimer XY. If
the corresponding decay constants are denoted respectively by
dX , dX2 , and dXY then the overall decay rate of X is given by
dX [X] + 2dX2 K X2 [X]2 + dXY K XY [X][Y], K XY being the rate constant
for the formation of the heterodimer XY (i.e., [XY] = K XY [X][Y]).
The analogous quantity for protein Y is dY [Y] + dXY K XY [X][Y].
5. By de¬nition, the steady-state concentration of each transcrip-
tion factor is determined by the solution of the equation that
results from setting its rate of synthesis equal to its rate of
68 BIOLOGICAL PHYSICS OF THE DEVELOPING EMBRYO




(c)

1

[X] < 0
[Y] < 0
[Y]
[Yo ]
2


3
[Y] = 0
[X] > 0
[X] = 0
[Y] > 0

[X]
[Xo]

Fig. 3.6 The solutions of the steady-state Eqs. 3.3 and 3.4, given in terms of the
steady-state solutions [X0 ] and [Y0 ]. Here [X0 ] is de¬ned as the steady-state cellular
level of monomer X produced in the presence of a steady-state cellular level [Y0 ] of
monomer Y, assuming a rate of transcription SX0 . Thus, by de¬nition (see Eq. 3.3),
SX0 = d X [X0 ] + 2d X2 K X2 [X0 ]2 + d XY K XY [X0 ][Y0 ]. Since along the red and blue lines,
r r
respectively, d[X]/dt ≡ [X] = 0 and d[Y]/dt ≡ [ Y] = 0, it is the intersections of these
curves that correspond to the steady-state solutions of the system of equations 3.3 and
3.4. Steady states 1 and 3 are stable, whereas steady state 2 is unstable.


decay. With the above ingredients, for the MRDH network one
arrives at

d[X]
= SXB ’ {dX [X] + 2dX2 K X2 [X]2 + dXY K XY [X][Y]} = 0, (3.3)
dt

1 + ρ K X K X2 [X]2
d[Y]
= SYB ’ dY [Y] + dXY K XY [X][Y] = 0. (3.4)
1 + K X K X2 [X]2
dt

Keller found that if in the absence of the repressor X the rate
of synthesis of protein Y is high then in its presence the system de-
scribed by Eqs. 3.3 and 3.4 exhibits three steady states, as shown in
Fig. 3.6. Steady states 1 and 3 are stable and thus could be considered
as de¬ning two distinct cell types, while steady state 2 is unstable.
In an example using realistic kinetic constants, the steady-state val-
ues of [X] and [Y] at the two stable steady states differ substantially
from one another, showing that the dynamical properties of these
autoregulatory networks of transcription factors can provide the basis
for generating stable alternative cell fates during early development.
The validity of Keller™s approach depends on the assumption that
the steady-state levels of transcription factors determine cell-fate
choice. In most cases this appears to be a reasonable assumption (Bate-
man, 1998; Becker et al., 2002). Sometimes, however (e.g., at certain
stages of early sea urchin development), transitions between cell types
3 CELL STATES: STABILITY, OSCILLATION, DIFFERENTIATION 69


appear to depend on the initial rates of synthesis of transcription
factors and not on their steady-state values (Bolouri and Davidson,
2003). Under conditions in which Keller™s assumption holds, however,
one can ask whether switching between these alternative states is
predicted to occur under realistic biological conditions. This can be
answered af¬rmatively: the external microenvironment of a cell that
contains an autoregulatory network of transcription factors could
readily induce changes in the rate of synthesis of one or more of
the factors via signal-transduction pathways that originate outside
the cell (˜˜outside--in signaling,” Giancotti and Ruoslahti, 1999). The
microenvironment can also affect the activity of transcription factors
in an autoregulatory network by indirectly interfering with their nu-
clear localization (Morisco et al., 2001). In addition, cell division may
perturb the cellular levels of autoregulatory transcription factors, par-
ticularly if they or their mRNAs are unequally partitioned between
the daughter cells. Any jump in concentration of one or more fac-
tors in the autoregulatory system can bring it into a new basin of
attraction and thereby lead to a new stable cell state.
In several well-studied cases, transcription factors regulating dif-
ferentiation in a mutually antagonistic fashion are initially co-
expressed in progenitors before one is upregulated and others
downregulated. Using a framework similar to Keller™s, Cinquin and
Demongeot (2005) have explored how known interactions among such
factors may give rise to multistable systems for cell-type determina-
tion.

Dependence of differentiation on cell“cell interaction: the
Kaneko“Yomo “isologous diversi¬cation” model
We have seen that the existence of multiple steady states in a cleaving
egg makes it possible, in principle, for more than one cell type to arise
among the descendents of such cells. However, this capability does
not, by itself, provide the conditions under which such a potentially
divergent cell population would actually be produced and persist as
long as it is needed. In later chapters we will see how the early embryo
and various organ primordia use chemical gradients and other short-
and long-range signaling processes to ensure that the appropriate cell
types arise at the correct positions to make a functional structure. In
certain cases (possibly including the early mammalian embryo) the
¬rst step in the developmental process may be to generate a mixture
of different cell types in a relatively haphazard fashion, just to get
things going. As we will see in the next chapter, sometimes it does not
matter what the initial arrangement of cells is -- differential af¬nity
can cause cells to ˜˜sort out” into distinct layers and lead to organized
structures from a randomly mixed population.
Consider the following observations: (i) during muscle differenti-
ation in the early Xenopus embryo, the muscle precursor cells must

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